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The Frobenius norm is an extension of the Euclidean norm to and comes from the Frobenius inner product on the space of all matrices. The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using Cauchy–Schwarz inequality.
The norm derived from this inner product is called the Frobenius norm, and it satisfies a submultiplicative property, as can be proven with the Cauchy–Schwarz inequality: [ ()] , if A and B are real matrices such that A B is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics.
A Frobenius matrix is a special kind of square matrix from numerical analysis. A matrix is a Frobenius matrix if it has the following three properties: all entries on the main diagonal are ones; the entries below the main diagonal of at most one column are arbitrary; every other entry is zero; The following matrix is an example.
where C is the companion matrix of the irreducible polynomial P, and U is a matrix whose sole nonzero entry is a 1 in the upper right-hand corner. For the case of a linear irreducible factor P = x − λ , these blocks are reduced to single entries C = λ and U = 1 and, one finds a ( transposed ) Jordan block.
One example is the squared Frobenius norm, which can be viewed as an -norm acting either entrywise, or on the singular values of the matrix: = ‖ ‖ = | | = =. In the multivariate case the effect of regularizing with the Frobenius norm is the same as the vector case; very complex models will have larger norms, and, thus, will be penalized ...
Hadamard product (matrices) Hilbert–Schmidt inner product; Kronecker product; Matrix analysis; Matrix multiplication; Matrix norm; Tensor product of Hilbert spaces – the Frobenius inner product is the special case where the vector spaces are finite-dimensional real or complex vector spaces with the usual Euclidean inner product
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
where [] is the augmented matrix with E and F side by side and ‖ ‖ is the Frobenius norm, the square root of the sum of the squares of all entries in a matrix and so equivalently the square root of the sum of squares of the lengths of the rows or columns of the matrix. This can be rewritten as